Is $x^{\frac{1}{2}}+ 2x+3=0$ a quadratic equation

Yes, but the quadratic is not in x.

$ 2 (\sqrt x )^2 + (\sqrt x ) + 3 = 0$ can be considered to be quadratic equation in $(\sqrt x ).$

That would not be considered quadratic in $x$, but you can let $u=\sqrt{x}$ to get $2u^2+u+3$. It WOULD be quadratic in $u$.